3.14.50 \(\int \frac {(2+3 x)^4}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=40 \[ -\frac {27 x^3}{10}-\frac {2079 x^2}{200}-\frac {21951 x}{1000}-\frac {2401}{176} \log (1-2 x)+\frac {\log (5 x+3)}{6875} \]

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {72} \begin {gather*} -\frac {27 x^3}{10}-\frac {2079 x^2}{200}-\frac {21951 x}{1000}-\frac {2401}{176} \log (1-2 x)+\frac {\log (5 x+3)}{6875} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-21951*x)/1000 - (2079*x^2)/200 - (27*x^3)/10 - (2401*Log[1 - 2*x])/176 + Log[3 + 5*x]/6875

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac {21951}{1000}-\frac {2079 x}{100}-\frac {81 x^2}{10}-\frac {2401}{88 (-1+2 x)}+\frac {1}{1375 (3+5 x)}\right ) \, dx\\ &=-\frac {21951 x}{1000}-\frac {2079 x^2}{200}-\frac {27 x^3}{10}-\frac {2401}{176} \log (1-2 x)+\frac {\log (3+5 x)}{6875}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.08 \begin {gather*} \frac {8 \log (-3 (5 x+3))-55 \left (2700 x^3+10395 x^2+21951 x+10814\right )}{55000}-\frac {2401}{176} \log (3-6 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-2401*Log[3 - 6*x])/176 + (-55*(10814 + 21951*x + 10395*x^2 + 2700*x^3) + 8*Log[-3*(3 + 5*x)])/55000

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^4}{(1-2 x) (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)),x]

[Out]

IntegrateAlgebraic[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)), x]

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fricas [A]  time = 0.62, size = 30, normalized size = 0.75 \begin {gather*} -\frac {27}{10} \, x^{3} - \frac {2079}{200} \, x^{2} - \frac {21951}{1000} \, x + \frac {1}{6875} \, \log \left (5 \, x + 3\right ) - \frac {2401}{176} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

-27/10*x^3 - 2079/200*x^2 - 21951/1000*x + 1/6875*log(5*x + 3) - 2401/176*log(2*x - 1)

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giac [A]  time = 0.93, size = 32, normalized size = 0.80 \begin {gather*} -\frac {27}{10} \, x^{3} - \frac {2079}{200} \, x^{2} - \frac {21951}{1000} \, x + \frac {1}{6875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {2401}{176} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-27/10*x^3 - 2079/200*x^2 - 21951/1000*x + 1/6875*log(abs(5*x + 3)) - 2401/176*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 31, normalized size = 0.78 \begin {gather*} -\frac {27 x^{3}}{10}-\frac {2079 x^{2}}{200}-\frac {21951 x}{1000}-\frac {2401 \ln \left (2 x -1\right )}{176}+\frac {\ln \left (5 x +3\right )}{6875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4/(1-2*x)/(5*x+3),x)

[Out]

-27/10*x^3-2079/200*x^2-21951/1000*x+1/6875*ln(5*x+3)-2401/176*ln(2*x-1)

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maxima [A]  time = 0.61, size = 30, normalized size = 0.75 \begin {gather*} -\frac {27}{10} \, x^{3} - \frac {2079}{200} \, x^{2} - \frac {21951}{1000} \, x + \frac {1}{6875} \, \log \left (5 \, x + 3\right ) - \frac {2401}{176} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

-27/10*x^3 - 2079/200*x^2 - 21951/1000*x + 1/6875*log(5*x + 3) - 2401/176*log(2*x - 1)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.65 \begin {gather*} \frac {\ln \left (x+\frac {3}{5}\right )}{6875}-\frac {2401\,\ln \left (x-\frac {1}{2}\right )}{176}-\frac {21951\,x}{1000}-\frac {2079\,x^2}{200}-\frac {27\,x^3}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 2)^4/((2*x - 1)*(5*x + 3)),x)

[Out]

log(x + 3/5)/6875 - (2401*log(x - 1/2))/176 - (21951*x)/1000 - (2079*x^2)/200 - (27*x^3)/10

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sympy [A]  time = 0.15, size = 36, normalized size = 0.90 \begin {gather*} - \frac {27 x^{3}}{10} - \frac {2079 x^{2}}{200} - \frac {21951 x}{1000} - \frac {2401 \log {\left (x - \frac {1}{2} \right )}}{176} + \frac {\log {\left (x + \frac {3}{5} \right )}}{6875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)/(3+5*x),x)

[Out]

-27*x**3/10 - 2079*x**2/200 - 21951*x/1000 - 2401*log(x - 1/2)/176 + log(x + 3/5)/6875

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